Symmetry properties of stable solutions of semilinear elliptic equations in unbounded domains

Abstract

We consider stable solutions of a semilinear elliptic equation with homogeneous Neumann boundary conditions. A classical result of Casten, Holland and Matano states that all stable solutions are constant in convex bounded domains. In this paper, we examine whether this result extends to unbounded convex domains. We give a positive answer for stable non-degenerate solutions, and for stable solutions if the domain $\Omega$ further satisfies $\Omega \cap \{\vert x\vert \le R\}= O(R^2)$, when $R\rightarrow +\infty$. If the domain is a straight cylinder, an additional natural assumption is needed. These results can be seen as an extension to more general domains of some results on De Giorgi’s conjecture. As an application, we establish asymptotic symmetries for stable solutions when the domain satisfies a geometric property asymptotically.

Appendices

Generalized principal eigenvalue

This section is devoted to defining the generalized principal eigenvalue of a linear operator, and to state some properties. The term generalized is used when dealing with unbounded domains, in which there may not exist eigenfunctions in $H^1$. Here, we focus on the essential aspects and omit the details: the content of this section will be developed in a forthcoming paper [45]

1.1 Definition

We generally consider a smooth domain $\Omega$ and a linear elliptic operator

$$\begin{aligned} \mathcal {L}u(x):=\mathrm {div}\left( A(x)\cdot \nabla u(x)\right) +B(x)\cdot \nabla u(x)+c(x)u(x),\quad \forall x\in \Omega , \end{aligned}$$

(21)

where, $c:\Omega \rightarrow \mathbb {R}$, $B:\Omega \rightarrow \in \mathbb {R}^n$, and $A: \Omega \rightarrow \mathbb {R}^{n\times n}$ such that A(x) is positive-definite (uniformly in $x\in \Omega$). For simplicity, we assume that the coefficients are smooth. We associate the operator $\mathcal {L}$ with Neumann boundary conditions

$$\begin{aligned} \mathcal {B} u(x):=\partial _{\nu _A} u(x)=0,\quad \forall x\in \partial \Omega , \end{aligned}$$

with $\nu$ the outer normal derivative and $\partial _{\nu _A} u:=\nu \cdot A\cdot \nabla u$ the co-normal outer derivative of u associated with A. We focus here on Neumann boundary conditions, but we keep the notation $\mathcal {B}$ to emphasize that our statements can be adapted to other boundary conditions.

We consider the following eigenproblem:

$$\begin{aligned} \left\{ \begin{aligned}&-\mathcal {L}\psi =\lambda \psi&\text {in }\Omega , \\&\mathcal {B} \psi =0&\text {on }\partial \Omega . \end{aligned}\right. \end{aligned}$$

(22)

If the domain is bounded, the Krein-Rutman theory gives the existence of an eigenvalue $\lambda _1$ to (22), called the principal eigenvalue. This eigenvalue is real and minimizes the real part of the spectrum. In addition, $\lambda _1$ is simple, and is the only eigenvalue associated with a positive eigenfunction (called principal eigenfunction). We let the reader refer to [31, 49] for more details. We point out that a fundamental property is that the validity of the Maximum Principle for the operator $(\mathcal {L},\mathcal {B})$ is equivalent to the condition $\lambda _1>0$.

If the domain is unbounded, Krein-Rutman’s theory cannot be applied because the elliptic operator does not have compact resolvents. However, we can still define the notion of principal eigenvalue. Following the approach of [11, 13, 51], we give the following definitions.

Definition A.1

• A function $u \in C^{2}({\overline{\Omega }})$ is said to be a subsolution (resp. supersolution) if

  $$\begin{aligned} \left\{ \begin{aligned}&-\mathcal {L}u \le 0 (\text {resp. }\ge 0)&\text {in }\Omega ,\\&\mathcal {B}u \le 0 (\text {resp. }\ge 0)&\text {on }\partial \Omega . \end{aligned}\right. \end{aligned}$$

• We define the generalized principal eigenvalue of $(\mathcal {L},\mathcal {B})$ as

  $$\begin{aligned} \lambda _1:=\sup \left\{ \lambda \in \mathbb {R}: (\mathcal {L}+\lambda ,\mathcal {B})\text { admits a positive supersolution} \right\} . \end{aligned}$$

  (23)

This definition coincides with the classical definition in the case of a bounded domain, and coincide with the definition (2) when $\mathcal {L}$ is self-adjoint. We are about to see that $\lambda _1$ admits a positive eigenfunction. However, $\lambda _1$ may not be simple.

1.2 Existence of a positive eigenfunction

A remarkable property is that, even in unbounded domains, $\lambda _1$ is associated with a positive eigenfunction.

Proposition A.2

Let $\Omega \subset \mathbb {R}^n$ be a smooth (possibly unbounded) domain and ${\mathcal {L}}$ an elliptic operator as in (21). There exists $\varphi \in C^2(\overline{\Omega })$ which is positive on $\overline{\Omega }$ and satisfies

$$\begin{aligned} \left\{ \begin{aligned}&-\mathcal {L}\varphi =\lambda _1 \varphi&\text {in }\Omega ,\\&\mathcal {B} \varphi =0&\text {on }\partial \Omega . \end{aligned}\right. \end{aligned}$$

(24)

We refer to $\varphi$ as a principal eigenfunction of $(\mathcal {L},\mathcal {B})$.

Proof

The proof follows closely [13] (see the proofs of Theorem 3.1 and Proposition 1). For any $R>0$, let $D_R\subset \mathbb {R}^n$ be a smooth connected open set such that $B_R\subset D_R\subset B_{2R}$, with $B_R$ the ball of radius R. We also choose $D_R$ to be increasing with R, and to be such that $\partial D_R\cap \partial \Omega$ is a $C^2$ $(n-2)$-dimensional manifold. Set $\Omega _R:=\Omega \cap D_R$ and consider the eigenvalue problem with mixed boundary conditions

$$\begin{aligned} \begin{aligned}&-\mathcal {L}\psi =\lambda \psi&\text {a.e. in }\Omega _R,\\&\mathcal {B} \psi =0&\text {a.e. on }\partial \Omega \cap D_R,\\&\psi =0&\text {a.e on }\Omega \cap \partial D_R. \end{aligned} \end{aligned}$$

From the results of Liberman [37], we know that all classical results (Schauder estimate, Maximum Principle, solvability, etc.) hold from the mixed boundary value problem above. As $\Omega _R$ is bounded, the weak Krein-Rutman theorem (e.g., Corollary 2.2 in [46]) provides a pair of principal eigenelements $(\lambda _1^{R},\varphi ^{R})$, where $\varphi ^R\in W^{2,n}$. From Hopf’s lemma, we have $\varphi >0$ on $\overline{\Omega }$. We choose the normalization $\varphi (0)=1$. Note that we impose Dirichlet boundary conditions on $\Omega \cap \partial D_R$ to ensure the decreasing monotonicity of $R\mapsto \lambda _1^{R}$. Hence, $\lambda _1^{R}$ converge to some ${\underline{\lambda }}_1$ when $R\rightarrow +\infty$.

Now, fix a compact $0\in K\subset {\overline{\Omega }}$ and assume that R is large enough so that $K\subset \overline{\Omega }_R\backslash D_R$. From Theorem 3.3 in [38] and Theorem 4.3 in [39], we derive a Harnack estimate, that is,

$$\begin{aligned} \sup _{K} \varphi ^R\le C\inf _K \varphi ^R \end{aligned}$$

with a constant C independent of R. From $\varphi (0)=1$, we deduce that $\varphi ^R$ is bounded in K, uniformly in R. From classical Schauder estimates, we deduce that $\varphi ^R$ is $C^{2,\alpha }(K)$, uniformly in R. Up to extraction of a subsequence, $\varphi ^{R}$ converges to some $\varphi$ in $C^2(K)$. From a diagonal argument, we are provided with $\varphi \in C^2(\overline{\Omega })$ which satisfies

$$\begin{aligned} \left\{ \begin{aligned}&-\mathcal {L}\varphi ={\underline{\lambda }}_1 \varphi&\text {in }\Omega ,\\&\mathcal {B} \varphi =0&\text {on }\partial \Omega , \end{aligned}\right. \end{aligned}$$

and $\varphi >0$ on $\overline{\Omega }$. Consequently $\lambda _1=\underline{\lambda _1}$, which achieves the proof. $\square$

As a direct consequence,

Corollary A.3

There exists a positive supersolution of $(\mathcal {L},\mathcal {B})$ in $\Omega$ if and only if $\lambda _1\ge 0$.

1.3 The Rayleigh–Ritz variational formula

In the self-adjoint case, i.e., if $B\equiv 0$ in (21), the principal eigenvalue can be expressed through the Rayleigh–Ritz variational formula. This result is classical in bounded domains.

Proposition A.4

Assume $\Omega$ is smooth (possibly unbounded) and that $\mathcal {L}$ is a self-adjoint elliptic operator. For $\lambda _1$ defined in (23), we have

$$\begin{aligned} \lambda _1=\inf \limits _{\begin{array}{c} \psi \in H^1(\Omega )\\ \Vert \psi \Vert _{{L}^2}=1 \end{array}} \mathcal {F}(\psi ) :=\inf \limits _{\begin{array}{c} \psi \in H^1(\Omega )\\ \Vert \psi \Vert _{{L}^2}=1 \end{array}}\int _\Omega \vert \nabla \psi \vert _A^2-c\psi ^2, \end{aligned}$$

(25)

with $\vert \nabla \psi \vert _A^2:=\nabla \psi \cdot A\cdot \nabla \psi .$

Note that since the coefficient c is bounded, from the dominated convergence theorem we deduce that the infimum in (25) can be taken equivalently on test functions $\psi \in C^1_0(\overline{\Omega })$.

Proof

From the dominated convergence theorem and classical density results, it is equivalent to take the infimum on compactly supported smooth test functions in (25), namely

$$\begin{aligned} \lambda _1=\inf \limits _{\begin{array}{c} \psi \in C^1_c({\overline{\Omega }})\\ \Vert \psi \Vert _{{L}^2}=1 \end{array}}\mathcal {F}(\psi ), \end{aligned}$$

where $C^1_c({\overline{\Omega }})$ is the space of continuously differentiable functions with compact support in ${\overline{\Omega }}$. The remaining of the proof is classical and can be adapted from the proof of Proposition 2.2 (iv) in [13] (which itself relies on [1, 11]). $\square$

1.4 The Maximum Principle

We give, as a complement, some results on the link between the sign of $\lambda _1$ and the validity of the Maximum Principle in unbounded domains. The proofs of the following statements are underlying in the content of the present article, and we leave the details to a forthcoming note [45].

Definition A.5

We say that $(\mathcal {L},\mathcal {B})$ satisfies the Maximum Principle if any subsolution with finite supremum is nonpositive.

For simplicity, we focus on self-adjoint operators, that is, we assume $B\equiv 0$ in (21). We can then express $\lambda _1$ through the Rayleigh–Ritz variational formula (25).

The first result states that the (strict) sign of $\lambda _1$ is equivalent to the validity of the Maximum Principle.

Proposition A.6

Assume $\mathcal {L}$ is self-adjoint.

• 1. If $\lambda _1>0$, $(\mathcal {L},\mathcal {B})$ satisfies the Maximum Principle.

• 2. If $\lambda _1<0$, $(\mathcal {L},\mathcal {B})$ does not satisfy the Maximum Principle.

No general answer holds for the degenerate case $\lambda _1=0$. Nevertheless, the following proposition states the validity of what could be called a Critical Maximum Principle when $\lambda _1\ge 0$ if the domain satisfies a growth condition at infinity.

Proposition A.7

Suppose that $\mathcal {L}$ is self-adjoint and that the domain $\Omega$ satisfies (5). Let $\varphi$ be an eigenfunction associated with $\lambda _1$. If $\lambda _1\ge 0$, then a subsolution with finite supremum is either nonpositive or a multiple of $\varphi$.

A first consequence of this result is the simplicity of $\lambda _1$ if it admits a bounded eigenfunction.

Corollary A.8

Under the same conditions, if $\lambda _1$ admits a bounded eigenfunction, then $\lambda _1$ is simple.

The simplicity of $\lambda _1$ should be understood as follows: if $\psi \in C^2({\overline{\Omega }})$ is a solution of (24), then it is a scalar multiple of $\varphi$.

We also give the following necessary and sufficient condition for the validity of the Maximum Principle in the critical case $\lambda _1=0$.

Corollary A.9

Under the same conditions, further assume $\lambda _1=0$, and let $\varphi$ be an associated eigenfunction. Then, $(\mathcal {L},\mathcal {B})$ satisfies the Maximum Principle if and only if $\varphi$ is not bounded.

On the different definitions of stability

When considering stability from a dynamical point of view, one can come up with the two following definitions.

Definition B.1

A solution u of (1) is said to be dynamically stable if, given any $\varepsilon >0$, there exists $\delta _0>0$ such that for any $v_0(x)$ with $\Vert v_0-u\Vert _{L^\infty }\le \delta _0$, we have

$$\begin{aligned} \Vert v(t,\cdot ) -u(\cdot )\Vert _{L^\infty }\le \varepsilon ,\quad \forall t>0, \end{aligned}$$

where v(t, x) is the solution of the evolution problem

$$\begin{aligned} \left\{ \begin{aligned}&\partial _tv(t,x)-\Delta v(t,x)=f(v(t,x))&\forall x\in \Omega ,\ \forall t>0,\\&\partial _\nu v(t,x)=0&\forall x\in \partial \Omega ,\ \forall t>0,\\&v(t=0,x)=v_0(x)&\forall x\in \Omega . \end{aligned}\right. \end{aligned}$$

(26)

Definition B.2

A solution u of (1) is said to be asymptotically stable if there exists $\delta _0>0$ such that for any $v_0(x)$ with $\Vert v_0-u\Vert _{L^\infty }\le \delta _0$, we have

$$\begin{aligned} \Vert v(t,\cdot ) -u(\cdot )\Vert _{L^\infty }\rightarrow 0,\quad \text {when }t\rightarrow +\infty , \end{aligned}$$

where v(t, x) is the solution of (26).

The following proposition clarifies the hierarchy of the different definitions of stability.

Proposition B.3

Let u be a solution of (1) and $\lambda _1$ from (2). The following implications hold

$$\begin{aligned} u \text { asymptotically stable}\Rightarrow u\text { dynamically stable}\Rightarrow \lambda _1\ge 0. \end{aligned}$$

Proof

The first implication is trivial. Let us show the second implication by contradiction: assume $\lambda _1<0$ and that u is dynamically stable. For $R>0$, define the truncated domain $\Omega _R:=\Omega \cap \{\vert x\vert <R\}$ and consider the following mixed-boundary eigenvalue problem: find $\lambda _{1,R}\in \mathbb {R}$ and $\varphi _R\in C^2(\Omega _R)$ satisfying

$$\begin{aligned} \left\{ \begin{aligned}&-\Delta \varphi _R-f'(u)\varphi _R=\lambda _{1,R} \varphi _R&\text {in }\Omega _R,\\&\partial _\nu \varphi _R=0&\text {on }\partial \Omega \cap \{\vert x\vert <R\},\\&\varphi _R=0&\text {on }\Omega \cap \{\vert x\vert =R\}. \end{aligned}\right. \end{aligned}$$

(27)

From a recent result of Rossi [51, Theorem 2.1], we know that, for almost every $R>0$, the eigenproblem (27) admits a unique eigenpair $(\lambda _{1,R},\varphi _R)$ such that $\varphi _R>0$ on $\overline{\Omega }\cap \{\vert x\vert <R\}$. Moreover, $R\mapsto \lambda _{1,R}$ is strictly decreasing and $\lim _{R\rightarrow +\infty }\lambda _{1,R}=\lambda _1$. Let us fix $R>0$ large enough such that $\lambda _{1,R}<0$. We also choose the normalization $\Vert \varphi _R\Vert _{L^\infty }=1$.

Let us consider a parameter $\varepsilon >0$ small enough such that

$$\begin{aligned} \eta _\varepsilon :=\sup \limits _{\begin{array}{c} {\tilde{u}}\in [\inf _{\Omega _R} u,\sup _{\Omega _R} u]\\ \vert h\vert \le \varepsilon \end{array}} \left| f'(\tilde{u})-\frac{f({\tilde{u}}+h)-f({\tilde{u}})}{h}\right| < -\lambda _{1,R}, \end{aligned}$$

(28)

and $\delta _0$ given by Definition B.1. Consider v the solution of the parabolic equation (26) with initial datum $v_0:=u+\delta _0\varphi _R$, and set $h(t,x)=v(t,x)-u(x)$. On the one hand, since $\Vert v_0-u\Vert _{L^\infty }\le \delta _0$, the stability assumption implies $\Vert h(t,\cdot )\Vert _{L^\infty }\le \varepsilon$ for all time $t\ge 0$. On the other hand, h satisfies

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t h(t,x)- \Delta h(t,x)\ge \left( f'(u(x))-\eta _\varepsilon \right) h(t,x)&\text {in }\Omega _R,\\&\partial _\nu h=0&\text {on }\partial \Omega \cap \{\vert x\vert <R\},\\&h\ge 0&\text {on }\Omega \cap \{\vert x\vert =R\} \end{aligned}\right. \end{aligned}$$

From the parabolic comparison principle, we infer $h(t,x)\ge \tilde{h}(t,x):= e^{-(\lambda _1+\eta _\varepsilon )t}\delta _0\varphi _R(x)$ for all $t\ge 0$ and $x\in \Omega _R$. Using that $\lambda _1+\eta _\delta <0$, we deduce that $\Vert h(t,\cdot )\Vert _{L^\infty }$ diverges to $+\infty$ when t becomes large: contradiction. $\square$

Remark B.4

Note that, in the proof, the perturbation $\delta _0 \varphi _R$ has a compact support in ${\overline{\Omega }}$ and an arbitrarily small $L^\infty$ norm. Thus, if $\lambda _1<0$ then (26) drives $u+h$ away from u for any h which is positive or negative on $\Omega _R$ if R is large enough.

One can ask whether the following implication holds:

$$\begin{aligned} \lambda _1 >0 \Rightarrow u\text { asymptotically stable}. \end{aligned}$$

(29)

This implication is classical when the domain is bounded (see Proposition 1.4.1 in [24]), but it is not clear whether it extends to unbounded domains.

Question 2

Does (29) hold in unbounded domains ?

We think that, in general, the answer is negative. Nevertheless, as a consequence of the results of the present paper, we give a positive answer for unbounded convex domains.

Proposition B.5

Let $\Omega \subset \mathbb {R}^n$ be a smooth convex domain (possibly unbounded) and u be a solution of (1). Then

$$\begin{aligned} \lambda _1>0\Rightarrow u \text { asymptotically stable}\Rightarrow u \text { dynamically stable}\Rightarrow \lambda _1\ge 0. \end{aligned}$$

Proof

From Proposition B.3, we only have to show the first implication. Assume $\lambda _1>0$. We deduce from Theorem 1.4 that u is constant. Thus $\lambda _1=-f'(u)$ and $\varphi$ is constant. We choose $\varepsilon$ small enough such that $\eta _{\varepsilon }\in (0,\frac{\lambda _1}{2})$ with $\eta _\varepsilon$ defined in (28). Let $v_0$ be as in Definition B.2 (we use the same notations). We set $T:=\sup \{t>0:\Vert h(t,\cdot )\Vert _{L^\infty }\le \varepsilon \}$. By continuity and the choice of $v_0$, we know that $T>0$. We have

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t h(t,x)-\Delta h(t,x)\le \left( f'(u)+\eta _\varepsilon \right) h(t,x)&\forall t\in (0,T),\ x\in \Omega ,\\&\partial _\nu h(t,x)=0&\forall t\in (0,T),\ x\in \partial \Omega . \end{aligned}\right. \end{aligned}$$

From $f'(u)+\eta _\varepsilon \le -\frac{\lambda _1}{2}$ and the parabolic comparison principle, we obtain $\Vert h(t,\cdot )\Vert _{L^\infty }\le \varepsilon e^{-\frac{\lambda _1}{2}t}$, for all $t\in (0,T)$. We deduce $T=+\infty$ and $\Vert h(t,\cdot )\Vert _{L^\infty }\rightarrow 0$ when $t\rightarrow +\infty$, thus u is asymptotically stable. $\square$

Isolation of stable solutions

We give a brief discussion on the isolation of stable solutions of (1) in the set of all solutions. Note that this question is crucial in the Proof of Theorems 1.6 and 4.1, since the key point is to show that $\Sigma _u$ is a discrete set. When considering the $L^\infty$ topology, we have the following.

Lemma C.1

Let $\Omega \subset \mathbb {R}^n$ be a smooth domain (possibly unbounded), and denote S the set of solutions of (1) in $\Omega$. Let $u\in S$ be either stable non-degenerate, or unstable non-degenerate (i.e. $\lambda _1<0$). Then, u is isolated in S for the $L^\infty (\Omega )$ topology.

This result is essentially classical, at least for bounded domains. We give a proof at the end of the section.

Note that, in general, this result fails for $L^\infty _{loc}$ topology. For example, consider the Allen–Cahn equation in $\mathbb {R}$, $-u''=u(1-u^2).$ This equation admits an explicit solution $u : x\mapsto \tanh \frac{x}{\sqrt{2}}$ which is stable (degenerate). On the one hand, the family of the translated solutions $u_a(\cdot )=u(\cdot -a)$ converges to 0 when $a\rightarrow +\infty$ in the $L^\infty _{loc}$ topology. On the other hand, 0 is a stable non-degenerate solution (because $f'(0)<0$).

One could argue that the above counterexample relies on the fact that the nonlinearity is balanced, that is, $\int _0^1 f=0$ with $f(u):=u(1-u)(u-\nicefrac {1}{2})$. However, one can build similar counterexamples for unbalanced nonlinearities by considering a ground state (which has been proved to exist in most cases, see, e.g., [10]).

In an attempt to extend the results of Sect. 4, we address the following question.

Question 3

Is $\Sigma _u$ from (20) always a singleton when u is stable non-degenerate?

We think the answer is negative; yet, we are not able to provide a counterexample.

Proof (Lemma C.1)

Assume that there exists $u_k\in S$, $u_{k+1}\not \equiv u_k$, a sequence which converges to u, and let us show that $\lambda _1=0$. We set $v_k:=u_{k+1}-u_k$. For all k, $u_k$ is a solution of (1) in $\Omega$, thus

$$\begin{aligned} \left\{ \begin{aligned}&-\Delta v_k-c_k(x)v_k=0&\text {in }\Omega ,\\&\partial _\nu v_k=0&\text {on }\partial \Omega , \end{aligned}\right. \end{aligned}$$

(30)

where

$$\begin{aligned} c_k(x):=\frac{f(u_{k+1}(x))-f(u_k(x))}{u_{k+1}(x)-u_k(x)}. \end{aligned}$$

Since f is $C^{1,\alpha }$ and $u_{k+1}-u_k$ is bounded, $c_k(x)$ converges uniformly to $f'(u(x))$ when $k\rightarrow +\infty$.

Formally, we have

$$\begin{aligned} \mathcal {F}\left( \frac{v_k}{\Vert v_k \Vert _{\begin{array}{c} {L}^2 \end{array}}}\right) \le \Vert f'(u)-c_k\Vert _\infty \underset{k\rightarrow 0}{\longrightarrow }0, \end{aligned}$$

(with ${\mathcal {F}}$ from (2)) which contradicts the fact that u is stable non-degenerate. However, the former calculation is not licit when $\Omega$ is unbounded. To make it rigorous, we use the cut-off function $\chi _R$ defined in (10).

Multiplying (30) by $v_k\chi _R^2$, integrating on $\Omega$, using the divergence theorem and the boundary condition in (30) we find

$$\begin{aligned} \mathcal {F}\left( \frac{\chi _Rv_k}{\Vert \chi _Rv_k\Vert _{\begin{array}{c} {L}^2 \end{array}}}\right)&=\frac{\int _{\begin{array}{c} \Omega \end{array}}\chi _R^2v_k^2 (c_k-f'(u))}{\int _{\begin{array}{c} \Omega \end{array}}\chi _R^2v_k^2}+\frac{\int _{\begin{array}{c} \Omega \end{array}} \vert \nabla \chi _R\vert ^2v_k^2}{\int _{\begin{array}{c} \Omega \end{array}}\chi _R^2v_k^2}\\&\le \Vert c_k-f'(u)\Vert _{L^\infty \left( \Omega _R\right) } +4\ \alpha _{k,R}, \end{aligned}$$

where

$$\begin{aligned} \mathcal {C}_k(R):=\int _{\Omega _R} v_k^2,\quad \alpha _{k,R}:= \frac{\mathcal {C}_k(2R)}{R^2\mathcal {C}_k(R)}. \end{aligned}$$

On the one hand, in the Proof of Theorem 1.4, we show that, for fixed $k\ge 0$, $\liminf \limits _{R\rightarrow +\infty } \alpha _{k,R}\le 0.$ On the other hand, since $\Vert c_k-f'(u)\Vert _{L^\infty \left( \Omega _R\right) }$ goes to 0 when $k\rightarrow +\infty$, uniformly in R, we deduce that $\mathcal {F}$ can be made arbitrarily small. It implies $\lambda _1\le 0$. The reverse inequality $\lambda _1\ge 0$ can be proved with the same method, which achieves the proof. $\square$